## $2$-Selmer groups of hyperelliptic curves with marked points

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- by Ananth N. Shankar PDF
- Trans. Amer. Math. Soc.
**372**(2019), 267-304 Request permission

## Abstract:

We consider the family of hyperelliptic curves over $\mathbb {Q}$ of fixed genus along with a marked rational Weierstrass point and a marked rational non-Weierstrass point. When these curves are ordered by height, we prove that the average Mordell–Weil rank of their Jacobians is bounded above by $5/2$, and that most such curves have only three rational points. We prove this by showing that the average rank of the $2$-Selmer groups is bounded above by $6$. We also consider another related family of curves and study the interplay between these two families. There is a family $\phi$ of isogenies between these two families, and we prove that the average size of the $\phi$-Selmer groups is exactly $2$.## References

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## Additional Information

**Ananth N. Shankar**- Affiliation: Department of Mathematics, Massachusetts Institute of Technology, Cambridge, Massachusetts
- Received by editor(s): December 15, 2017
- Received by editor(s) in revised form: December 16, 2017, and February 21, 2018
- Published electronically: October 10, 2018
- © Copyright 2018 American Mathematical Society
- Journal: Trans. Amer. Math. Soc.
**372**(2019), 267-304 - MSC (2010): Primary 11G10; Secondary 11G30, 14G05
- DOI: https://doi.org/10.1090/tran/7546
- MathSciNet review: 3968769